Authors need help doing the Math

[Solfe]

I write a blog on Dungeons and Dragons. This game uses a variety of dice to generate numbers. I am trying to think my way through a posting about generating an number using the sum of 3 six sided dice. Obviously, the chance of rolling any number on one die is 16.67% or 1 in 6, the best totalled roll on 3 six sided dice is 18 and the worst is 3.

Last year, I posted about the probability of get certain numbers on two paired dice in a different game. The game valued higher rolls as better, but doubles also had special meaning. This roll was compared to a list of outcomes, parts of a story, which were not mathematical in nature but could be generalized as good, bad, and in between. That obscured some game mechanics, which actually made it easier to get better results than you would otherwise think.

I would like to do a blog post on the probability spread of manipulated dice. A lot of people just make up stuff when selecting the method to roll this number that should be in a range of 3 to 18, but is skewed towards 18. Usually they say things like "roll 4 six sided dice and discard the lowest result" or "reroll any 1's". Usually these rules for skewing the die rolls are not mathematically based, but just something the referee thought sounded good at that point in time. Sometimes the rules are dynamic, where the referee says "reroll any result of 1 on a die" but really means "reroll the first 1 you roll, but all other results of 1 stand". Can you see how these rules are not selected based on probability? The skewing is different depending on what the ruling is. I want to evaluate the difference between these methods using probability.

Does anyone have a suggestion for what happens when you make up a convention for rolling dice in a skewed fashion, or perhaps a website or book?

[grant hutchison]

Any basic book on probability will give you what you want, which is an approach to identifying the individual probabilities involved, and then multiplying them together. But each random referee ruling is going to require different maths, so there's not really a rule of thumb you can apply--just a succession of specific scenarios.

Grant Hutchison

[Solfe]

Any basic book on probability will give you what you want, which is an approach to identifying the individual probabilities involved, and then multiplying them together. But each random referee ruling is going to require different maths, so there's not really a rule of thumb you can apply--just a succession of specific scenarios.

Grant Hutchison

Yes, that was what I was thinking. It's funny but the only book I have on probability is actually a book on Quantum Mechanics. It is less than helpful in this case. I just want die results, not electron orbitals.

[Noclevername]

Yes, that was what I was thinking. It's funny but the only book I have on probability is actually a book on Quantum Mechanics.

"You enter the Dungeon. A goblin both does and does not attack you...."

[grant hutchison]

Yes, that was what I was thinking. It's funny but the only book I have on probability is actually a book on Quantum Mechanics. It is less than helpful in this case. I just want die results, not electron orbitals.

So any basic school text-book devoted to probability. Mine are fifty years old, so probably not worth mentioning their titles.

ETA: Actually, I'm now recalling being told, by a member here, that high schools in the USA don't even address calculus in any useful detail. So make that an introductory textbook for university undegraduates. (Not that you need calculus, but an absence of basic calculus from high-school curricula suggests to me that they might not be addressing probability to any useful extent for your purposes.)

Grant Hutchison

[Noclevername]

The way I play the game, a DM basing events on what makes the best gameplay, is more important than calculating specific probabilities to produce true mathematical randomness.

Both I and other DMs I've seen will even occasionally "fudge" the results of a roll if the results would impede the roleplaying and storytelling aspects. Usually when TPK (total party kill) might occur in a way the PCs are helpless to prevent; Rocks Fall Everyone Dies. Occasional frustration and character death are part of role-playing, but taking away all agency and forcing them to roll a whole new party only frustrates players. The game must be fun above all, or it's not worth doing.

[profloater]

With three fair dice abc every number is one in 6.6.6 = 216 but there are six combinations with the same outcome,
Abc,acb,bac,bca,cab,cba, so any pattern is a permutation achieved six ways, assuming the dice are equally important, but the order does not matter.
To score 3 the same is unique, 1,1,1 all three have that 1/6 chance so the probability is 1/216

Same with 666 but the numbers in between have several solutions such as 9 which can be made:
126,135,144,225,234,333,
If we take 126, 135,144 the chance of the first 1 is 1/6
But after the one, the second die can be 2,3,4 which is 3/6. The third die is then fixed and the chance is 1/6.
That makes 1/6*3/6*1/6= 1/72
But the order does not matter so we have 6/72 or 1/12.
Then we have 225 and 234.
To get 2 is 1/6 but the second can be 2 or 3, 2/6.
But again the third has to be right 1/6,
We have 1/6*2/6*1/6 and again six combinations so 1/18.
Finally 333 is 1/216
So the chance of score 9 is 1/12+1/18+1/216

Every score can be worked out that way.
The point is a number like 333 can only happen one way but 123 can happen 6 ways. To score 4 requires 112, which can be done 3 ways, so the chance is 3/216, that applies to combinationa aab, (aab,aba,baa) I hope i got that right!

[profloater]

So the chance of each score out of 216 is
18 or 3,1;
17 or 4,3;
16 or 5,6;
15 or 6,10;
14 or 7,15;
13 or 8,21;
12 or 9,25;
11 or 10,27;

Total 216 as a check.

[profloater]

Maybe worth noting, normally the probability p of an independent event occurring in n trials is found in a useful formula

Probability of r hits with probability p each of n trials is nCr.p^r.(1-p)^(n-r)
NCr is n!/r!(n-r)!.

But dice are tricky when the scores are added as shown previously.
However the symmetrical scores are similar to the binomial series.

[Solfe]

The way I play the game, a DM basing events on what makes the best gameplay, is more important than calculating specific probabilities to produce true mathematical randomness.

Both I and other DMs I've seen will even occasionally "fudge" the results of a roll if the results would impede the roleplaying and storytelling aspects. Usually when TPK (total party kill) might occur in a way the PCs are helpless to prevent; Rocks Fall Everyone Dies. Occasional frustration and character death are part of role-playing, but taking away all agency and forcing them to roll a whole new party only frustrates players. The game must be fun above all, or it's not worth doing.

I fully agree with fudging die rolls at the table, but I want to figure out how these different methods of stat generation actually change the distribution of scores. In some cases, I think people are picking a method without actually knowing what it does.

For example, sometime a stat or a situation will impart a modifier of 4 on a 20 sided die. Without foreknowledge of such things, you have to modify a preplanned scenario on the fly which kind of defeats the purpose of preplanning. In fact, the DM may run of out time or ideas by the time he realizes what happened. By spelling out how these different methods alter probability in the planning phase, a DM can get ahead of the curve.

I got in a comedic situation in a game where my character had ridiculously high bonus with a bow but by the same standard he was easy to hit. My solution was to hide in a house with another character to open and close a door as I shot. At first, the DM tried to make it a challenge by making me roll dice to see if could time my shots to the door opening. Since timing relied on the same stat as shooting, it wasn't a challenge at all. Amusingly, the DM's second attempt to change the situation was more pragmatic. Someone grabbed the door knob to prevent characters from opening the door, which didn't require any die rolls. The characters were too afraid of the creature outside to fight it at close range and barricaded the door, which took chance out of the equation. The outcome was a place the DM didn't expect to be and didn't have any plans for it. If only he had known about those ridiculously high and low stats.

[profloater]

So do you need a method for any nuber of sides?

[profloater]

Rereading, to skew the dice scores by rules like that! It is straightforward to calculate for a given rule like reroll the 1s but some rules would introduce condions so that rolls are not individual.
Like roll one die, then roll as many dice as that number. Sounds like spreadsheet fun.

[21st Century Schizoid Man]

"Skew" has a very specific meaning in probability, which may not be exactly what you mean here.

What exactly do you want? Do you want to be able to determine the full probability distribution (that is, all the possible outcomes and their probabilities) of some of the schemes? Or are summary statistics (like the average value) enough?

If you roll four and throw the lowest one away, the average value is 15869/1296, which is about 12.245. If you just roll three and keep them all, the average is 10.5.

[Solfe]

"Skew" has a very specific meaning in probability, which may not be exactly what you mean here.

What exactly do you want? Do you want to be able to determine the full probability distribution (that is, all the possible outcomes and their probabilities) of some of the schemes? Or are summary statistics (like the average value) enough?

If you roll four and throw the lowest one away, the average value is 15869/1296, which is about 12.245. If you just roll three and keep them all, the average is 10.5.

True. I am probably thinking of both a "generic skewing" where someone sets out to get higher die rolls via a mechanic like eliminating outliers, but also the "probabilistic skew" of having outliers be a factor. A single character has 6 stats made up of the sum of 3 (or 4) rolls of the dice for 6 different stats. I don't think you could say much about probabilistic skew on the basis of rolling one die 3 times, but when you do that dozens of times it comes in to play.

What made me think of this was a mechanic: "Discard rolls of one". which I took at face value until I rolled 4 ones in a row. On reading the text further, it was ambiguous because the next sentence said to discard "the one". It seemed like the author (not me) assumed that you'd would or could only roll a one once per 4 die rolls. Obviously he knew the average of 3.5 on a six sider and determined it was not likely to do it twice in a row, but didn't account for physically rolling a die 4 times.

So I am thinking of what both concepts mean.